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points -- produces the ideal and initial ideal from the coordinates of a finite set of points

Synopsis

Description

This function uses the Buchberger-Moeller algorithm to compute a grobner basis for the ideal of a finite number of points in affine space. Here is a simple example.
i1 : M = random(ZZ^3, ZZ^5)

o1 = | 6 7 8 3 0 |
     | 6 7 0 8 8 |
     | 1 8 6 9 1 |

              3        5
o1 : Matrix ZZ  <--- ZZ
i2 : R = QQ[x,y,z]

o2 = R

o2 : PolynomialRing
i3 : (Q,inG,G) = points(M,R)

                    2                     2        2   3          131 2  
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - ---z  -
                                                                  106    
     ------------------------------------------------------------------------
     700    2683    5607    14717        389 2   1563    2940    45553   
     ---x - ----y + ----z + -----, x*z + ---z  - ----x - ----y - -----z +
     583     583    1166     583         106      583     583     1166   
     ------------------------------------------------------------------------
     44157   2    35 2   1404    3950    4903    3453         29 2   5052   
     -----, y  + ---z  + ----x - ----y - ----z - ----, x*y - ---z  - ----x -
      583        106      583     583    1166     583        106      583   
     ------------------------------------------------------------------------
     4662    3481    35715   2   215 2   3785    861    22123    16767   3  
     ----y + ----z + -----, x  + ---z  - ----x - ---y - -----z + -----, z  -
      583    1166     583        106      583    583     1166     583       
     ------------------------------------------------------------------------
     939 2   840    2520    49922    60336
     ---z  + ---x + ----y + -----z - -----})
      53     583     583     583      583

o3 : Sequence
i4 : monomialIdeal G == inG

o4 = true

Next a larger example that shows that the Buchberger-Moeller algorithm in points may be faster than the alternative method using the intersection of the ideals for each point.

i5 : R = ZZ/32003[vars(0..4), MonomialOrder=>Lex]

o5 = R

o5 : PolynomialRing
i6 : M = random(ZZ^5, ZZ^150)

o6 = | 5 0 1 4 5 5 7 0 4 1 7 5 4 9 8 1 7 7 7 7 7 7 8 3 4 6 8 8 0 3 7 8 2 0 8
     | 4 8 6 1 5 0 0 0 8 1 2 2 7 2 9 6 5 1 4 5 7 0 8 4 2 2 0 5 4 5 7 6 8 9 5
     | 7 2 7 4 5 2 7 1 3 3 5 8 3 4 8 8 1 4 7 0 4 4 6 3 2 1 6 5 9 7 1 0 9 7 9
     | 7 9 8 9 5 2 2 6 1 2 9 1 0 4 4 1 1 3 6 7 7 5 4 8 8 4 6 7 3 8 1 6 4 1 5
     | 1 2 5 9 3 7 6 4 0 6 0 1 5 4 4 8 0 9 1 3 3 6 9 2 7 6 0 3 5 4 0 7 1 2 5
     ------------------------------------------------------------------------
     2 4 7 0 8 5 5 6 1 5 7 7 3 5 7 3 2 6 6 7 3 3 9 3 7 3 3 5 2 8 0 0 2 2 3 6
     2 5 2 6 9 4 5 5 9 1 7 1 5 4 2 3 4 1 0 0 5 0 5 3 9 1 1 0 6 8 5 3 9 7 8 5
     2 9 4 0 3 6 2 0 8 0 5 7 6 5 1 4 9 3 8 7 9 6 8 0 5 3 7 2 4 2 2 7 4 8 6 5
     9 4 3 8 9 2 8 4 5 5 9 3 8 3 3 3 6 0 1 6 0 0 2 6 2 8 4 3 5 1 0 1 1 2 8 2
     2 4 9 5 0 5 1 5 6 1 5 2 6 3 7 1 0 1 3 1 1 5 5 3 7 9 3 9 7 9 9 6 4 7 3 3
     ------------------------------------------------------------------------
     8 9 4 8 9 0 6 0 6 5 0 7 2 9 2 5 1 8 2 8 3 2 5 0 9 1 7 1 4 4 2 1 9 1 4 6
     7 9 4 8 4 8 0 9 5 7 8 9 5 7 6 7 7 9 8 3 5 6 3 2 7 3 5 1 8 8 8 2 2 4 7 9
     0 3 7 5 7 9 9 4 6 7 4 9 7 7 9 2 5 1 6 6 3 7 3 0 0 3 8 7 4 6 0 2 2 7 8 4
     4 6 5 4 1 3 5 1 4 6 1 2 9 2 4 6 6 6 6 2 3 9 5 6 0 5 0 8 4 9 3 5 9 7 5 0
     6 7 7 8 1 9 2 9 2 7 0 6 0 8 2 2 7 6 7 9 1 1 8 0 7 5 9 1 3 3 2 4 1 7 0 1
     ------------------------------------------------------------------------
     3 9 3 7 0 3 4 6 2 5 2 6 9 5 7 0 6 3 6 8 2 4 7 9 4 9 9 9 1 9 2 7 2 5 9 3
     9 9 6 9 7 4 3 7 0 0 6 6 5 9 9 1 9 8 3 3 2 6 4 3 0 1 0 2 3 0 0 5 2 0 7 8
     7 1 7 7 3 8 9 6 2 8 8 9 2 6 9 1 5 5 6 8 9 6 9 6 0 8 2 3 1 5 5 5 6 2 5 1
     1 8 1 7 8 5 7 5 6 8 6 8 8 3 9 0 8 9 1 5 4 2 1 7 8 7 7 7 0 6 7 1 6 7 7 7
     5 5 2 2 2 7 8 4 8 7 2 1 5 3 6 7 0 2 5 2 9 4 1 2 2 2 3 8 1 3 1 2 8 7 3 8
     ------------------------------------------------------------------------
     9 6 8 0 8 3 6 |
     3 1 8 0 3 4 3 |
     6 7 8 3 7 7 3 |
     4 1 7 7 3 9 3 |
     8 6 7 7 6 8 2 |

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 6.50841 seconds
i8 : time C = points(M,R);
     -- used 0.64004 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :